It remains to prove assertion (iv). Statements pertaining to foliations of \(W^{s/u}_{loc}(S_{ode,\varepsilon })\)  follows directly from Fenichel theory [1]}, [2]}, [3]}, [4]}. Those pertaining to foliations of \(W^{s/u}_{loc}(S_{\varepsilon ,h})\)  for fixed \(h\)  follow from Theorem REF . To see that the bounds on the contraction rate \(\chi _A(h)\)  are given by (REF ), notice that by (REF ) we have
\(|\mu _{a,j}(z)| = | 1 + h \lambda _{a,j,ode}(z) | = 1 + h \textrm {Re}\, \lambda _{a,j,ode}(z) + O(h^2) ,\) 
